xiaoya wang, fan shao, xiaogong hu, hao yang

Xiaoya Wang, Fan Shao, Xiaogong Hu, Hao Yang

Shanghai Astronomical Observatory, Chinese Academy of Sciences,

Shanghai 200030, China

Contents:

1. Introduction

2. SHAO SLR SINEX solution

3. SLR Intra-technique Combination

4. Results and analysis

5. Conclusion

1.Introduction

As we know, Satellite Laser Ranging(SLR) play an importantrole for the determination of Terrestrial Reference Frame(TRF)and EOP. It is only technique for the origin of TRF or ITRF andworks together with Very Long Baseline Interferometry(VLBI) todetermine the scale factor.

Datum definition

Origin Defined by SLR

Scale Defined by SLR and VLBI

Orientation Align to ITRF2008’s orientation(core sites)

Table. Datum definition of ITRF

1.IntroductionNow, there are 7 ILRS Analysis Centers(ACs). Because of the

different models, strategies, methods and other reasons used in thedata processing, their products are not completely same. Which isthe best? They are often evaluated by combination. In general, thecombination solution is the best one. The solution closer to thecombination solution is better.

Analysis Centers(AC)

ASI Italian Space Agency

BKG Bundesamt für Kartographie und Geodäesie

GFZ Helmholtz Centre Potsdam German Research Centre for Geosciences

DGFI Deutsches Geodätisches Forschungsinstitut

JCET Joint Center for Earth System Technology

NSGF NERC Space Geodesy Facility

GRGS Groupe de Recherche de Géodésie Spatiale

Table. List of 7 ACs

1.Introduction

Now, there are 2 combination centers(CCs). They useweekly SINEX solutions from each AC as input and generateILRS combination solution ILRSA and ILRSB. Which combinationsolution is better?

ASI(main)

JCET(back up)

Every AC’s SINEX

Every AC’s SINEX

ILRSB

ILRSAcombination

combination

1.Introduction

ShangHai astronomy observatory provided SLR products to IERSsince 1985 and became associate AC since 1999, providing global SLRdata quality report including time and range bias and navigationsatellite orbit to ILRS weekly. However, weekly SINEX solutions andcombination products with other ACs have not been provided. So, weinvestigate our SINEX solutions and the combination solutions withother SLR ACs. Our SINEX solutions is named SHAO and our combinedsolutions is named ILRSC for comparison with other ACs and CCs. Bycomparison we have analyzed the accuracy and stability of ourproducts.

As associate AC , SHAO provided BIAS report with residual RMSabout 1cm.

2. SHAO SLR SINEX solution2.1 Processing model of SHAO SINEX solution

MODELS TRF ITRF2014 as a prior coordinate

Troposphere Mendes-Pavlis zenith delay model

COM ILRS station dependent CoM model

Relativity point-mass accelerations, Lense-Thirring effect, Coriolis

force

Precession IERS2010

Nutation IAU 2006+IERS

Geopotential EGM2008 (static terms 70x70, C(2,0), C(2,1) ,S(2,1) time

dependent)

Tidal forces solid earth tides: IERS 2010 Conventions model

Ocean tides: FES2004

Third-body ephemeris: JPL DE421

ESTIMATED

PARAMETERSStation

coordinate

a priori values: SLRF2014/1.0m sigma constraints

EOP a priori values: IERS 14 C04/xp,yp :20 mas ;lod:2 ms sigma

Range bias Estimation for some non core station, sigma:1.0m

2. SHAO SLR SINEX solution2.2 SHAO SINEX solutions

3-D coordinate residuals w.r.t SLRF2014(Lageos1/2 + Etalon1/2)

2. SHAO SLR SINEX solution2.2 SHAO SINEX solutions

EOP residuals w.r.t EOP C04

(Lageos1/2 + Etalon1/2)

EOP accuracyXp: 0.11±0.21 masYp: 0.08±0.17 mas LOD: 0.01±0.06 ms

3.SLR Intra-technique Combination

3.1 Processing of a priori constraints

As for the constraints of station coordinates and EOP,there are two methods to deal with the a prioriconstraints for SLR intra-technique combination. First isstraightforward method that the combined SINEXsolution is directly based on SINEX solutions from variousACs due to the loosely constrained solutions. Another isminimal constraints method. We firstly remove the looseconstraints of provided SINEX solutions and then imposeminimal constraints on them. Finally, the SLR combinedweekly solution is obtained with minimal constraint aswell.


3.1 Processing of a priori constraints

method advantages disadvantages

straightforward

method

convenient in calculation; a

priori information is not

required

orientation is random

minimal constraints

method

orientation unified as

certain reference frame

causing rank deficient of

normal equation when

remove loose constraints

Table. Comparison of two methods for a priori constraint processing


3.2 Determination of relative weight factors

There are mainly two approaches to determine the relative weight factors in SLR intra – technique combination.

Determination of weight factors

1. restraint condition iteration

2. variance component estimation



The first approach is the iterative calculation of weightfactors based on the formulas (1) and (2). This method isassuming that the final combined residual 𝜒2 = 1 and thecontribution of every AC to the combined residual is the same.The weight factors of every AC is determined throughcontinuous iteration and the termination condition of iteration is:

𝑅1𝑇(𝜎1Σ1)

−1𝑅1 = ⋯ = 𝑅𝑛𝑇(𝜎𝑛Σ1)

−1𝑅𝑛 (1)

𝜒2 = 𝑅1𝑇Σ1

−1𝑅1 +⋯+ 𝑅𝑖𝑇Σ𝑖

−1𝑅𝑖 = 1 (2)



Another method for relative weight factors is based onvariance component estimation. In this method, the initialweight factors of each AC are set to 1 and the newiteration’s weight matrix is determined by multiply theweight factor from last iteration by the weight matrix.When the difference value between two successiveiteration is less than 0.001, the iteration is stopped.

𝜀𝑖 =𝑣𝑖𝑇𝑃𝑖𝑣𝑖

𝑛𝑖−𝑡𝑟(𝑁−1𝐴𝑖

𝑇𝑃𝑖𝐴𝑖)


3.3 ILRSC intra-technique combination method

Here, we list the methods of ILRSA, ILRSB and our combination products named ILRSC

combination

method

processing of a priori

constraints

determination of weight

factors

ILRSA straightforward method restraint condition iteration

ILRSB minimal constraints method variance component

estimation

ILRSC straightforward method variance component

estimation

Table. Different methods of every combined solution

4.Results and analysis

Based on the method mentioned above, we combined the SINEX solutions that 8 ILRS ACs provide during Jan 1 1993 and Dec 31, 2017. The combined solution is analyzed by comparing our products with ILRS corresponding products. The analysis is performed according to the following aspects:

① Relative weight factors of each AC② Accuracy analysis of station coordinates and EOP③ Analysis of translation parameters and scale factor④ Difference between SLRF2008 and SLRF2014⑤ Evaluation of SHAO SINEX solutions


4.1 Relative weight factors of each AC

combined solution ASI BKG DGFI ESA GFZ GRGS JCET NSGF

ILRSA 1.00 1.51 4.06 1.20 1.45 1.80 2.03 2.44

ILRSB 1.00 1.07 2.06 1.16 1.93 1.22 1.36 2.36

ILRSC 1.00 1.13 3.19 1.05 1.94 1.10 1.61 2.17

Table/Figure the variance factors comparison of each ACs in ILRSC combination weekly solution

ASI performs the best and then ESA, GRGS, BKG, JCET, GFZ, NSGF, DGFI (ASI is supposed as unit weight 1.00 )


4.2 Accuracy analysis of station coordinates and EOP

To assess the accuracy of the combination station coordinates andEOP the weekly combined solution is compared with SLRF2008 andIERS EOP C04. We use HELMERT seven parameters to transformweekly reference frame to SLRF2008, the transformationrelationships are as follows:

𝑋𝑟𝑖(𝑡0) = 𝑋𝑤

𝑖 − 𝑡𝑗 − 𝑡0 ∙ 𝑋 𝑟𝑖 + 𝑇1 + 𝐷 ∙ 𝑋𝑤

𝑖 − 𝑅3𝑌𝑤𝑖 + 𝑅2𝑍𝑤

𝑖

𝑌𝑟𝑖(𝑡0) = 𝑌𝑤

𝑖 − 𝑡𝑗 − 𝑡0 ∙ 𝑌𝑟𝑖 + 𝑇2 + 𝐷 ∙ 𝑌𝑤

𝑖 − 𝑅1𝑍𝑤𝑖 + 𝑅3𝑋𝑤

𝑖

𝑍𝑟𝑖 (𝑡0) = 𝑍𝑤

𝑖 − 𝑡𝑗 − 𝑡0 ∙ 𝑍 𝑟𝑖 + 𝑇3 + 𝐷 ∙ 𝑍𝑤

𝑖 + 𝑅1𝑌𝑤𝑖 + 𝑅2𝑋𝑤

𝑖

(4

𝑋𝑟𝑃 = 𝑋𝑤

𝑝+ 𝑅2

𝑌𝑟𝑃 = 𝑌𝑤

𝑝+ 𝑅1

𝑙𝑜𝑑𝑟 = 𝑙𝑜𝑑𝑤


4.2 Accuracy analysis of station coordinates and EOPTime series of weekly 3-Dresiduals for Xp, Yp, LODwith respect to EOP C04are given for individualAC solutions and thecombined ILRSC solutionafter transformation.The results show that

polar motion of Xp is0.1875mas, Yp is0.1759mas and LOD is0.0485ms. The accuracyof combined solution isobvious better than thatof individual AC solution.It’s almost the same asILRSA.

4.Results and analysis4.2 Accuracy analysis of station coordinates and EOP

Figure. Time series of weekly 3-D residuals with respect to SLRF 2008 for ILRS core sites areobtained for individual AC solution and the combined ILRSC solution. The results show that3D accuracy of combined station coordinates is 4.33mm

4.Results and analysis4.2 Accuracy analysis of station coordinates and EOP

POS-3D(mm) Xp(mas) Yp(mas) LOD(ms)

ASI 7.61(±5.29) -0.034(±0.243) -0.011(±0.229) -0.006(±0.062)

BKG 10.01(±7.38) -0.041(±0.254) 0.012(±0.240) -0.002(±0.071)

DGFI 10.71(±6.89) 0.026(±0.254) -0.046(±0.247) 0.001(±0.074)

ESA 10.67(±6.89) -0.014(±0.240) 0.025(±0.217) -0.009(±0.085)

GFZ 9.42(±6.60) -0.015(±0.288) 0.008(±0.279) -0.012(±0.139)

GRGS 7.84(±5.46) -0.040(±0.236) 0.006(±0.226) -0.001(±0.062)

JCET 10.18(±9.00) -0.046(±0.238) 0.006(±0.224) -0.002(±0.055)

NSGF 9.20(±5.03) -0.015(±0.303) 0.001(±0.291) -0.035(±0.189)

ILRSA 5.51(±4.38) 0.012(±0.209) -0.004(±0.202) -0.002(±0.048)

ILRSB 5.43(±4.73) * * *

ILRSC 5.67(±4.33) -0.035(±0.187) 0.002(±0.176) -0.001(±0.048)

4.Results and analysis4.3 Analysis of translation parameters and scale factor

we use HELMERT 7 parameters to transform the combined weekly solutions to SLRF2008. Three combined solutions show consistency to some extent. But from 1993 to 2014, ILRSC translation parameters are more consistent with ILRSA. From 2005 to 2017, scale parameters are more identical with ILRSB. The reason is not clear. But this means the existence of the third CC is necessary.


Figure. Time series comparison of the scale parameter of our combined solution ,ILRSA

and ILRSB with respect to SLRF2008


Tx(mm) Ty(mm) Tz(mm) Scale(ppb)

ILRSA 0.63(±3.80) 0.82(±3.46) -1.08(±6.57) 0.85(±0.62)

ILRSB 0.75(±4.38) 0.78(±3.85) -1.51(±8.63) 0.64(±0.63)

ILRSC 0.43(±3.76) 0.94(±3.55) -0.99(±6.40) 0.79(±0.62)

From the table we can see that the mean values of ILRSC translation parameters and scale parameters are close to those of ILRSA and ILRAB. But the standard deviations are little smaller and more stable w.r.t ILRSA and ILRSB.


we detected thecharacteristic of origin andscale parameter of the ILRSC.By fitting dominant linearterms of translation and scaleparameters, we get linearchange rate of translationparameters and scaleparameters are 0.0330mm/yr,0.0969mm/yr, 0.3345mm/yrand 0.0438ppb/yr. As shown inthe figure, the origins show alinear change mainly in Zdirection.


4.3 Analysis of translation parameters and scale factor

After theselinear change areremoved theFourier spectrumanalysis is appliedto them. We findthe translationparameters havean annual term of2.564mm in X,2.556mm in Y and3.466mm in Z.

4.Results and analysis4.4 Analysis of difference between SLRF2008 and SLRF2014

we can see thatamplitude of translationparameters of SLRF2014is obvious smaller w.r.tSLRF2008 after 2014.The change curve ismore smooth both fortranslation parametersand the scale factor. Thisshows SLRF2014 is morestable than SLRF2008


4.4 Analysis of difference between SLRF2008 and SLRF2014

Linear fitting results oftranslation and scale parametersof ILRSC combined weeklysolution w.r.t SLRF2014 showsthe linear change rate oftranslation and scale parametersare 0.0342mm/yr, 0.0388mm/yr,0.0584mm/yr and 0.0106ppb/yr.The translation and scaleparameters’ change rate in Xdirection is little higher than thatof SLRF2008. But in Y and Zdirection they show a significantreduction than that of SLRF2008 .


4.5 Evaluation of SHAO SINEX solutions

Table. SHAO SINEX solution relative weight factor comparison with other ACs

AC ASI BKG DGFI ESA GFZ GRGS JCET NSGF SHAO

mean 8.85 11.43 20.21 11.67 14.97 10.5 11.75 10.45 11.19

Std

deviation15.37 22.15 23.04 15.66 17.78 15.57 21.77 18.86 18.66



Figure. Translation parameters w.r.t SLRF2014



Figure. Scale parameters w.r.t SLRF2014

5. Conclusion

◆ SHAO could provide SLR SINEX solutions with accuracyabout 1cm 3D coordinates and 0.11±0.21 mas for Xp,0.08±0.17 mas for Yp and 0.01±0.06 ms for LOD. Itsaccuracy and stability are the middle of ILRS ACs

◆ SHAO also could provide the combination solutionswith 5.67±4.33 mm for 3D coordinates and -0.035±0.187 mas for Xp, 0.002 ±0.176 mas for Yp and -0.001±0.048 ms for LOD.

◆ ILRSC relative weight factors are consistent with thoseof ILRSA and ILRSB. They show the same results forbad solutions at average level. ILRSC stationcoordinates and EOP are better than that of individualAC too.

5. Conclusion

◆ ILRSC translation and scale parameters showconsistency with that of ILRSA and ILRSB. From1993 to 2004, these parameters are moreconsistent with that of ILRSA, but from 2005 to2017, they are more consistent with that of ILRSB.The above comparison results verify the reliabilityof ILRSC combination solutions and also show theexistence of the third CC is specially necessary.

◆ The stability of TRF is discussed by ILRSC combinedweekly solutions w.r.t SLRF2008 and SLRF2014respectivily. From the characteristic comparisons oftranslation and scale parameters, we can verifythat SLRF2014 is more stable than SLRF2008.

Thank you for your attention！

AcknowledgmentsIERS/ILRS, ILRS ACs and ILRS CCs

the National Key R&D Program

(No.: 2016YFB0501405)

National Natural Science Foundation of China

(No.: 11973073)

xiaoya wang, fan shao, xiaogong hu, hao yang

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